REVIEW
The Pole Mass of The Heavy Quark. Perturbation Theory and Beyond
High Energy Physics - Phenomenology · Submitted 27 Feb 1994 · 41 references
Pith's verdict: The paper that made the pole mass unusable as a precision input, and the core argument still holds up.
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Referee report
The paper
The paper claims
The heavy quark pole mass carries an irreducible $O(50)$ MeV ambiguity, the paper argues, because an infrared renormalon makes it undefinable beyond perturbation theory.
The reading
What carries the argument
The load-bearing object is the infrared renormalon in the pole mass, exposed through the static-limit integral $100$ MeV, where the $100$ MeV pole of the gluon propagator is the only surviving piece and the integral becomes linearly divergent in the infrared. Summing bubble chains replaces $100$ MeV by the running coupling, producing the series $100$ MeV with $100$ MeV, whose Borel transform has a pole at $100$ MeV; the associated irreducible ambiguity is $100$ MeV. The paper's second piece of machinery is the three-dimensional character of the operator product expansion below $100$ MeV: because all energy transfers to the static quark vanish, only spatial momenta are separated, and the resulting linear sensitivity has no four-dimensional local operator to land on. The third piece is the mixing of $100$ MeV with $100$ MeV at one loop, which is what forces any HQET definition of the residual mass, and hence of $100$ MeV, to carry an explicit normalization point.
Extended reading notes
Core claim
The paper's central claim is that the pole mass of a heavy quark has no precise definition in the full theory. The argument runs on two independent tracks that are made to agree. First, a direct one-loop calculation with an infrared cutoff $100$ MeV shows that the pole mass differs from the mass normalized at that cutoff by a term linear in the cutoff, $100$ MeV — the classical Coulomb self-energy of a static colour source, which survives because the static limit forces the gluon energy $k_0$ to zero and reduces the four-dimensional integral to a three-dimensional one. Second, dressing the gluon line with a chain of vacuum-polarization bubbles and replacing the coupling by the running $100$ MeV turns that same linear term into a divergent series whose coefficients grow as $100$ MeV, giving a Borel-plane singularity at $100$ MeV and a best-possible truncation error of order $100$ MeV. The two routes give the same answer, and a renormalization-group argument for $100$ MeV reproduces it a third time. The authors stress what does not follow: the linear term cannot be written as the matrix element of a local operator, because the only candidate of the right dimension, $100$ MeV, is reduced by the equations of motion to operators that can only generate $100$ MeV corrections. The physical reason is that fixing the pole mass requires an infinitely long time, so the relevant object is a non-local expectation value along an open temporal Wilson line, not a condensate. Consequently the standard operator-product-expansion remedy — cut off the infrared, absorb the remainder into a gluon condensate, cancel the cutoff dependence — is unavailable here, and the pole mass should be kept out of any analysis that aims at $100$ MeV accuracy. The paper then shows by explicit one-gluon diagrams that the same factorial divergence appearing in the radiative corrections to the semileptonic width cancels against the one in the mass, order by order in the soft region, so that the width is genuinely free of the linear ambiguity when written in terms of the running mass.
Load-bearing premise
The derivation assumes that inserting the running coupling into the static integrand — justified in QED by the Ward identity but not automatic in non-abelian covariant gauges — faithfully captures the leading infrared behaviour, so that the linear term is a real property of the theory rather than an artifact of the cutoff prescription.
Editorial extensions
If this is right
- Extractions of $100$ MeV from the absolute semileptonic widths of $B$ and $D$ mesons lose their theoretical meaning; what such fits determine is $100$ MeV, the running mass at its own scale.
- The HQET parameter $100$ MeV must be replaced by a running $100$ MeV whose scale dependence is governed by $100$ MeV, so the familiar number $100$ MeV is scheme- and scale-dependent rather than a fixed constant.
- The requirement that the residual mass of HQET vanish cannot define the pole mass, because the operator mixing that generates it reappears at any other normalization point.
- The cancellation between the mass renormalon and the width's radiative corrections means the inclusive semileptonic width can be organized as an OPE in $100$ MeV with corrections starting only at $100$ MeV, so no linear $100$ MeV term contaminates it.
- Lifetime differences between hadrons of the same heavy flavour, which go as $100$ MeV, remain clean predictions, in contrast to their mass differences, which go as $100$ MeV.
Reading between the lines
- If the linear ambiguity is real, then any short-distance mass definition differing from the pole mass by an $100$ MeV renormalon ambiguity — the $100$ MeV mass included — inherits a residual sensitivity to scales below its own normalization point, so 'renormalon-free' mass schemes are a substantive design constraint rather than a bookkeeping choice.
- The same static-limit mechanism should generate linear ambiguities in any quantity that requires an infinite time to define, such as exclusive transition form factors away from zero recoil, which the authors flag as an open question rather than a result.
- The numerical coefficient of the linear term is scheme-dependent while its presence is not, so a two-loop calculation of the renormalon residue would turn the paper's rough $100$ MeV estimate into a falsifiable number comparable with lattice determinations of the same residue.
- Because the ambiguity is non-local, it suggests that the right non-perturbative input for the pole mass is a temporal Wilson-line correlator rather than a local condensate, an object that lattice QCD could in principle compute directly.
Original abstract
The key quantity of the heavy quark theory is the quark mass $m_Q$. Since quarks are unobservable one can suggest different definitions of $m_Q$. One of the most popular choices is the pole quark mass routinely used in perturbative calculations and in some analyses based on heavy quark expansions. We show that no precise definition of the pole mass can be given in the full theory once non-perturbative effects are included. Any definition of this quantity suffers from an intrinsic uncertainty of order $\Lam /m_Q$. This fact is succinctly described by the existence of an infrared renormalon generating a factorial divergence in the high-order coefficients of the $\alpha_s$ series; the corresponding singularity in the Borel plane is situated at $2\pi /b$. A peculiar feature is that this renormalon is not associated with the matrix element of a local operator. The difference $\La \equiv M_{H_Q}-m_Q^{pole}$ can still be defined in Heavy Quark Effective Theory, but only at the price of introducing an explicit dependence on a normalization point $\mu$: $\La (\mu )$. Fortunately the pole mass $m_Q(0)$ {\em per se} does not appear in calculable observable quantities.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The key quantity of the heavy quark theory is the quark mass $m_Q$. Since quarks are unobservable one can suggest different definitions of $m_Q$. One of the most popular choices is the pole quark mass routinely used in perturbative calculations and in some analyses based on heavy quark expansions. We show that no precise definition of the pole mass can be given in the full theory once non-perturbative effects are included. Any definition of this quantity suffers from an intrinsic uncertainty of order $\Lam /m_Q$. This fact is succinctly described by the existence of an infrared renormalon generating a factorial divergence in the high-order coefficients of the $\alpha_s$ series; the corresponding singularity in the Borel plane is situated at $2\pi /b$. A peculiar feature is that this renormalon is not associated with the matrix element of a local operator. The difference $\La \equiv M_{H_Q}-m_Q^{pole}$ can still be defined in Heavy Quark Effective Theory, but only at the price of introducing an explicit dependence on a normalization point $\mu$: $\La (\mu )$. Fortunately the pole mass $m_Q(0)$ {\em per se} does not appear in calculable observable quantities.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The key quantity of the heavy quark theory is the quark mass $m_Q$. | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| s1 | Since quarks are unobservable one can suggest different definitions of $m_Q$. | background 1.00 | 0.07 | 0.02 | hedged possibility 0.90 |
| s2 | One of the most popular choices is the pole quark mass routinely used in perturbative calculations and in some analyses based on heavy quark expansions. | background 1.00 | 0.04 | 0.02 | asserted 0.99 |
| s3 | We show that no precise definition of the pole mass can be given in the full theory once non-perturbative effects are included. | result 0.98 | 0.85 | 0.80 | asserted 1.00 |
| s4 | Any definition of this quantity suffers from an intrinsic uncertainty of order $\Lam /m_Q$. | result 0.98 | 0.65 | 0.43 | asserted 1.00 |
| s5 | This fact is succinctly described by the existence of an infrared renormalon generating a factorial divergence in the high-order coefficients of the $\alpha_s$ series; the corresponding singularity in the Borel plane is situated at $2\pi /b$. | interpretation 0.80 | 0.26 | 0.15 | asserted 1.00 |
| s6 | A peculiar feature is that this renormalon is not associated with the matrix element of a local operator. | result 0.62 | 0.56 | 0.05 | asserted 1.00 |
| s7 | The difference $\La \equiv M_{H_Q}-m_Q^{pole}$ can still be defined in Heavy Quark Effective Theory, but only at the price of introducing an explicit dependence on a normalization point $\mu$: $\La (\mu )$. | result 0.86 | 0.45 | 0.10 | asserted 0.98 |
| s8 | Fortunately the pole mass $m_Q(0)$ {\em per se} does not appear in calculable observable quantities. | interpretation 0.87 | 0.53 | 0.26 | asserted 0.98 |
What was examined, and what was not
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Figures
Showing 2 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9402360 by the authors.
References
How the literature uses it
92 arXiv papers cite this in 124 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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The pole mass of heavy quarks is {\em not} well-defined on the {\em non}-perturbative level -- i.e., it is {\em not} Borel summable in total QCD.
Papers citing it · 102 sentences without a confident label
Uses its method or tool 1 paper
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We use the $1S$ scheme [ref] to achieve cancellation of leading renormalon ambiguities from the mass parameter $$ against those in the factorially-growing coefficients of the $$ perturbative power series [ref] .
Appraising constrained second-order power corrections in HQET with $\Lambda_b \to \Lambda_c l \nu$ · $1S$ scheme and numerical inputs · 1.00
Uses its data or values 1 paper
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For this case the large-$β_0$ approximation fixes $C_0$ in the $MS$ scheme as [ref] C_0=-C_F e^5/6 , where $C_F=4/3$ for QCD.
Flavor dependence of normalization constant for an infrared renormalon · Introduction · 0.95
Compares with it 3 papers
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They are well constrained by the measurements of charm and beauty production in DIS and the fitted values (see Table [ref] in the Appendix [ref]) are consistent with the ones obtained in the corresponding HERA analyses [ref] within the intrinsic theoretical systematic uncertainty of the pole mass definition [ref] .
Impact of heavy-flavour production cross sections measured by the LHCb experiment on parton distribution functions at low x · Details of the QCD analysis · 0.96
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The presence of renormalons in leptonic observables seems to be in contrast with what is found in inclusive semileptonic decays of heavy flavours [ref] .
All-orders behaviour and renormalons in top-mass observables · Conclusions · 0.98
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Note that the overall amplitude $C_F e^5/6/π$ here is the same as that for the mass renormalon series in the large $β_0$ limit [ref] .
Effects of threshold resummation for large-$x$ PDF in large momentum effective theory · The leading renormalon series · 0.84
Cites it as support 3 papers
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And we aim to demonstrate that this contribution fully captures the leading IR-renormalon singularity observed in the perturbative pole mass definition [ref] .
Trace-anomaly-subtracted $\sigma$-mass for Heavy Quarks and Matching from Lattice QCD · Introduction · 0.96
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It was pointed out first in 1994 that the pole mass is not well-defined at the non-perturbative level [ref] .
The Dynamics of Beauty & Charm Hadrons and top quarks in the Era of the LHCb & Belle II and ATLAS/CMS, Motto: Non-perturbative QCD \& Many-body Final States · Non-perturbative Renormalons · 0.97
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Although less obvious, this is also the case for the factor $Γ(t→W b)$ (see Refs. [ref] ).
All-orders behaviour and renormalons in top-mass observables · Inclusive cross section without cuts · 0.99
Cites it as background 14 papers
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While the pole mass $$ suffers from the renormalon ambiguity, an infrared sensitivity of the order of the scale of quantum chromodynamics (QCD) [ref] , short-distance masses e.g. the $$ mass $(μ_m)$ and the MSR mass $(R)$ [ref] do not.
Single-differential top quark pair production cross sections with running mass schemes at NLO · Introduction · 0.88
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The results are already known since Refs. [ref] .
On the Light Massive Flavor Dependence of the Large Order Asymptotic Behavior and the Ambiguity of the Pole Mass · Asymptotic High Order Behavior and Borel Transform for Massless Lighter Quarks · 1.00
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This ambiguity, often called the “renormalon” ambiguity [ref] , is related to the growth of the coefficients, which is factorial.
More on minimal renormalon subtraction · Statement of the problem · 0.90
Discussion
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